We define non-pluripolar products of an arbitrary number of closed positive (1, 1)-currents on a compact Kähler manifold X. Given a big (1, 1)-cohomology class α on X (i.e. a class that can be represented by a strictly positive current) and a positive measure µ on X of total mass equal to the volume of α and putting no mass on pluripolar sets, we show that µ can be written in a unique way as the top degree self-intersection in the nonpluripolar sense of a closed positive current in α. We then extend Kolodziedj's approach to sup-norm estimates to show that the solution has minimal singularities in the sense of Demailly if µ has L 1+ε -density with respect to Lebesgue measure. If µ is smooth and positive everywhere, we prove that T is smooth on the ample locus of α provided α is nef. Using a fixed point theorem we finally explain how to construct singular Kähler-Einstein volume forms with minimal singularities on varieties of general type.

We study degenerate complex Monge-Ampère equations of the form (ω + dd c ϕ) n = e tϕ µ where ω is a big semi-positive form on a compact Kähler manifold X of dimension n, t ∈ R + , and µ = f ω n is a positive measure with density f ∈ L p (X, ω n ), p > 1. We prove the existence and unicity of bounded ω-plurisubharmonic solutions. We also prove that the solution is continuous under a further technical condition.In case X is projective and ω = ψ * ω ′ , where ψ : X → V is a proper birational morphism to a normal projective variety, [ω ′ ] ∈ N S R (V ) is an ample class and µ has only algebraic singularities, we prove that the solution is smooth in the regular locus of the equation.We use these results to construct singular Kähler-Einstein metrics of non-positive curvature on projective klt pairs, in particular on canonical models of algebraic varieties of general type.

International audienc

We study degenerate complex Monge-Ampère equations on a compact Kähler manifold (X, ω). We show that the complex Monge-Ampère operator (ω + dd c ·) n is well defined on the class E(X, ω) of ω-plurisubharmonic functions with finite weighted Monge-Ampère energy. The class E(X, ω) is the largest class of ω-psh functions on which the Monge-Ampère operator is well defined and the comparison principle is valid. It contains several functions whose gradient is not square integrable. We give a complete description of the range of the operator (ω + dd c ·) n on E(X, ω), as well as on some of its subclasses. We also study uniqueness properties, extending Calabi's result to this unbounded and degenerate situation, and we give applications to complex dynamics and to the existence of singular Kähler-Einstein metrics.